In what sense does the Laplace Transform give components of a signal in something called the 's-domain'? For example, I understand how this make sense for the Fourier transform. When we do the transform, we get 'how much of each frequency' in present in the signal. We get the value of the coefficients, such that when we multiply each coefficient with the corresponding constant frequency complex exponential and add those up, we get back our original signal. When engineers analyse the graph of amplitude squared in the frequency domain, they get the idea of how much of each frequency is present.
Is the same true for the Laplace transform as well? I know that Laplace transform is about getting "components" of a signal in terms of decaying complex exponentials, as opposed to unit magnitude complex exponentials in Fourier transform.
I haven't read much in detail about the inverse of Laplace transform. But its formula does not seem like we're summing up 'components times decaying exponential' over the entire s-plane. So what do we see when we look at, say, a colored graph in the 's-domain'? Is it really giving us the strength of the signal at each point in the s-plane? How is that true if we are not summing over the entire s-plane to get back our signal? Is this question clear?
 A: From my answer to a similar question:
You just have to accept the simple formulas belonging to the Laplace transform. Unlike the Fourier transform of functions or signals the Laplace transform of transient phenomena $t\mapsto f(t)$ $\>(t\geq0)$ has no intuitive physical interpretation. It is a purely formal operation applied to given or as yet unknown function terms ("expressions") resulting in other function terms hopefully listed in a catalogue. – While functions $f$ available only in the form of a discrete time series are Fourier transformed all the time, and interesting information about $f$ is revealed in this way,  nobody would consider Laplace transforming such a data set, nor would anybody look at the graph of a Laplace transform. 
A: Laplace Transform is a generalization of Fourier transform in the sense it can handle a much wider applications in Engineering, Pure and Applied Math.  The big difference it can handle signals which grows in time such as step, ramp or quadratic etc. Another feature of Laplace transform is it can readily solve (Initial Value Problem (IVP) while yield Fourier transform for steady state solution as a special case when s lies on the jω axis.
Recent advances in Control theory is in large part thanks to Laplace transform.  Bode, Nyquist plot are just tools engineers used daily in their design work.  It turns out, as far as system stability is concerned, one only to dig into the  Laplace transform X(s) in the s domain instead of the Fourier Transform X(jω).  For example, Laplace transform is widely used in Aerospace applications such as aircrafts, missiles and satellites.
Thanks to the availability of GPS, satellite/wireless communication and the Internet, their growing importance and influence in our daily life  make Cell phones a "must have toy" and account for a lot of popularity among young people as well as grown up.
The basic tool of Laplace transform when mathematicians, physicists and engineers alike realizes they are not just a tool for analysis but also for design. H∞ design, Poles placement Controller Design, Regulator and Observer Design, Stability Analysis are the latest examples of how powerful the tools have become.  As adaptive control and nonlinear systems gain popularity, Laplace transforms remains hot, as it offers the first clue on how nonlinear system will behave when we linearize the system around its operating point
using Taylor's Series approximation.
Needless to say, Fourier transform retains its major role in areas such as image and speech signal processing and machine vision. As we enter the age of Robotics, Self driving Cars, Artificial intelligence, it will be interesting to see who are the ultimate winners.  I suspect both will.  So will our mathematicians, physicists, engineers.  Doctors, drug researchers and developers, medical researchers. have made significant inroad in advancing gene therapy to control cancer or even cure disease.  Hopefully, everybody benefits from the progress and effort we made.
